We've looked at both arithmetic sequences and geometric sequences; let's wrap things up by exploring recursive sequences.

Recursion is the process of starting with an element and performing a specific process to obtain the next term.

If we go with that definition of a recursive sequence, then both arithmetic sequences and geometric sequences are also recursive.

Why? In an arithmetic sequence, each term is obtained by adding a specific number to the previous term.

In a geometric sequence, each term is obtained by multiplying the previous term by a specific number.

If a sequence is recursive, we can write recursive equations for the sequence. Recursive equations usually come in pairs: the first equation tells us what the first term is, and the second equation tells us how to get the n^th term in relation to the previous term (or terms).

Example 1: Write recursive equations for the sequence 5, 7, 9, 11,...

Solution: The first term of the sequence is 5, and each term is 2 more than the previous term, so our equations are:

a₁ = 5
a_n = a_n-1 + 2, for n > 1

Notice that we had to specify n > 1, because if n = 1, there is no previous term!

Example 2: Write recursive equations for the sequence 2, 4, 8, 16,...

Solution: The first term is 2, and each term after that is twice the previous term, so the equations are:

a₁ = 2
a_n = 2a_n-1, for n > 1

Example 3: Write recursive equations for the sequence 1, 1, 2, 3, 5, 8, 13, ...

Solution: This sequence is called the Fibonacci Sequence. Each term is the sum of the two previous terms. Since our recursion involves two previous terms, we need to specify the value of the first two terms:

a₁ = 1
a₂ = 1
a_n = a_n-1 + a_n-2, for n > 2

Example 4: Write recursive equations for the sequence 2, 3, 6, 18, 108, 1944, 209952,...

Solution The terms of this sequence are getting large very quickly, which suggests that we may be using either multiplication or exponents. It turns out that each term is the product of the two previous terms. Since our recursion uses the two previous terms, our recursive formulas must specify the first two terms.

a₁ = 2
a₂ = 3
a_n = (a_n-1)(a_n-2), for n > 2